Proceedings of the International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, 19-21 March, 2008, Hong Kong
Direction of Arrival Estimation Using Polynomial Roots Intersection for Multi-Dimensional Estimation (PRIME) H. K. Hwang, Zekeriya Aliyazicioglu, Marshall Grice, Anatoly Yakovlev, Peter Lu
Abstract— An array antenna system with innovative signal processing can enhance the resolution of a signal direction of arrival (DOA) estimation. Super resolution algorithms take advantage of array antenna structures to better process the incoming signals. They also have the ability to identify multiple targets. This paper explores the Polynomial Roots Intersection for Multi-dimensional Estimation (PRIME) algorithms.
applications in interference rejection [1], electronic steering [2], multiple beam forming [3], etc. This technology is now widely used in communication, radar, sonar, seismology, radio astronomy, etc. Consider a general array of M sensors as shown in Figure 1. The coordinate of the ith sensor is ri = [xi, yi, zi]T, i = 0, 1, … , M-1
The PRIME algorithm allows a polynomial rooting approach to estimate joint azimuth/elevation parameters of the signal detection for planer arrays. This method calculates a finite set of root intersections, which are the simultaneous solutions from multiple independent multivariate polynomials. The solutions for the source angle information are included in the simultaneous solutions. The PRIME algorithm does not require the use of a scan vector to scan all possible directions. The results demonstrate improvement in both resolution and computational efficiency with no loss of accuracy.
z
Ideal Signal DOA
3 4
Index Terms— DOA estimation, antenna arrays, array signal processing. I. SENSOR ARRAY SYSTEM A. Sensor Output Signal Model An array sensor system has multiple sensors distributed in space. This array configuration provides spatial sampling of the received waveform. A sensor array has better performance than the single sensor in signal reception and parameter estimation. Its superior spatial resolution provides a means to estimate the DOA of multiple signals. A sensor array also has This work was supported in part by the Raytheon H. K. Hwang and Zekeriya Aliyazicioglu are with the Electrical and Computer Engineering Department, California State Polytechnic University, Pomona, CA 91768 USA, (e-mail: hkhwang@ csupomona.edu and
[email protected]). Marshall Grice was a graduate student at California State Polytechnic University, Pomona, CA. He is now with the Boeing, CA. (e-mail:
[email protected]) Anatoly Yakovlev was an undergraduate student at California State Polytechnic University, Pomona, CA. He is now with Western Digital, San Jose, CA 95138, USA (e-mail:
[email protected]) Peter Lu is an undergraduate student at California State Polytechnic University, Pomona, CA. is now with the Raytheon, CA. (e-mail:
[email protected])
ISBN: 978-988-17012-1-3
α
θ
Calculated Signal DOA
2 y
0 5
Statistical analysis of the performance of the processing algorithm and processing resource requirements are discussed in this paper. Extensive computer simulations are used to show the performance of the algorithms.
ri
1
ϕ
6
sensor
x
Figure 1 An Array of M Sensors
Suppose a plane target signal waveform comes from the direction of k = [sinθcosϕ, sinθsinϕ, cosθ]T, where θ is the elevation angle and ϕ is the azimuth angle. The difference of the propagation path of this wave between the origin and the ith sensor ∆di is ∆di = riT k= sinθ(xicosϕ + yisinϕ) + zicosθ
(1)
where ri = [xi, yi, zi]T is the coordinate vector of the ith element. The corresponding propagation time delay τi is τi = ∆di /c
(2)
where c is the speed of light. If the bandwidth of signals is sufficiently narrow, then the data picked up by different sensor elements are related by a pure phase factor. The relative phase shift of the ith sensor with respect to the reference sensor at the origin is βi =
2π T 2π ri k = − sinθ(x i cosφ + y i sinφ ) λ λ
(3)
To avoid the effect of grating lobes, the distance between the two neighbor sensors has to be no more than one half of the wavelength. If the reference sensor is located at the
IMECS 2008
Proceedings of the International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, 19-21 March, 2008, Hong Kong origin, and a waveform received by the reference sensor due to signal coming from direction of k is x(t), then the received waveform at ith sensor is xi(t) = x(t-τi).
antenna is shown in Figure 2. Antenna elements are assumed to be placed on x-y plane and inter-element spacing d equals a half wavelength.
For the sensor array with M elements, we can define the array input vector x(t) and the array weighting vector w as x(t) = [x0(t), x1(t), ... , xM-1(t)]T
(4)
w = [w0, w1, … , wM-1]T
(5)
y
x d=λ/2
*
where xi(t) is the data input to the ith sensor and w i is the weight of the ith sensor. The sensor array output y(t) is H
y(t) = w x(t)
(6)
where the superscript represents the complex conjugate transpose (Hermitian) of the matrix. Suppose there are L independent signal sources impinging the antenna and we want to use a sensor array system to identify their directions of arrival (DOA). The input signal to each individual sensor is the combination of L independent signals. Every sensor in the array also receives random environmental ambient noise. This noise is modeled as Additive White Gaussian Noise (AWGN). The input waveform of the ith sensor element xi(t) is given by L
xi(t) =
∑s k =1
k,i
(t) + ni(t), i = 0,1, . . ., M-1
(7)
where sk,i(t) = sk,o(t-τk,i), and sk,o(t) denote the kth signal picked up by the sensor at the origin. ni(t) is the noise at ith sensor, τk,i is the relative delay of kth signal at the ith sensor. For the narrowband input signals, signal sk,i(t) is related to the signal sk,o(t) by a phase shift factor of βk,i. If the input signals have a wide bandwidth, the delay time of the signal at the ith sensor from reference signal at the origin may not be an integer multiple of the sampling time; additional interpolation filtering is required to emulate their delay. The weighted sum of samples of all sensors forms the array output. To estimate the DOA of wideband signals, each single weight is replaced by a tape delay lines filter. Such a processor is referred to as the Space Time Adaptive Processor (STAP) [4]. In this report, only the narrowband signal is considered. DOA estimation of conventional eigen-analysis methods uses a scan vector to scan over all possible directions and signal directions corresponding to the peak of the power spectrum. A more efficient method of estimating the DOA of signal is based on the root algorithm. A signal’s DOA can be determined by finding the roots of the characteristic polynomial whose amplitudes are closest to unity. The two dimensional array antenna studied in this report consists of 7 elements arranged in a honeycomb configuration. The geometric configuration of this array
ISBN: 978-988-17012-1-3
Figure.2 Seven Element Array Antenna
B. Narrowband Signal Simulations Simulation of the narrowband signal is relatively straightforward. The narrowband signal is defined as the signal bandwidth is a small fraction of c/D where c is the speed of light and D is the length of one dimensional array or the diameter of the two dimensional array. The narrowband signal s(t) can be expressed as s(t) = m(t) e
j2πf c t
(8)
where fc is the center frequency of the narrowband signal. Due to the propagation path difference, the signal at the ith sensor si(t) is related to the reference so(t) by si(t) = so(t–τi) = m(t–τi) e
j2πf c (t − τ i )
(9)
where τi = ∆di/c is the relative delay of the ith sensor and reference sensor. For the narrowband signal, m(t–τi) ≈ m(t), thus si(t) = so(t) e Equation (3).
jβ i
where the phase factor βi is given by
If there are L signals impinging on this array, assume the kth signal at the reference sensor is sk,0(t). For the narrowband waveform, this signal at the ith sensor is related to the reference signal sk,0(t) by a phase factor βk,i as shown as si(t) = sk,o(t) e
jβ k,i
.
The continuous sensor output waveform may be sampled at the sampling rate. Define the received data vector at sample n as x(n) = [x0(n), x1(n), … , xM-1(n),]T, where xk(n), k = 0, 1, … , M-1 is the signal at kth sensor and x0(n) is the signal at the reference sensor. The received data vector x(n) consists of a signal component s(n) and a noise component n(n), x(n) = s(n) + n(n) where L
s(n) =
∑s k =1
k,0
(n) v k
(10)
where vk is the array manifold vector of kth signal and
IMECS 2008
Proceedings of the International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, 19-21 March, 2008, Hong Kong A. Computer Simulations
⎡ 1 ⎤ ⎢ e jβ k,1 ⎥ ⎥ vk = ⎢ ⎢ # ⎥ ⎢ jβ k,M-1 ⎥ ⎣e ⎦ e
jβ k, i
(11)
The array elements are defined in Figure 3. If the sensors are ordered starting with the origin in the x-y plane then moving from the positive x axis clockwise The phase vector a of this array is given by a = [1 z2 zw z-1w z—2 z-1w-1 zw-1]T
is phase factor of ith element due to kth signal.
⎡ n 0 (n) ⎤ ⎢ n (n) ⎥ 1 ⎥ n(n) = ⎢ ⎢ : ⎥ ⎢ ⎥ ⎣n M −1 (n)⎦
(16)
zw−1
z −1w−1 6
7
(12) 5 z
−2
and ni(n), i = 0, 1, … , M-1 are independent white noise sequences.
1
z2
1
4
2
3 zw
z −1w
II. POLYNOMIAL ROOT INTERSECTION FOR MULTI-DIMENSIONAL ESTIMATION
Figure 3 Array Phase as a Function of z and
For a narrowband signal, if the jth signal’s DOA is (θj, ϕj), the relative phase shift of kth element due to the jth signal is defined by the following Equation:
Assume there is only one narrowband signal impinging on this array from angle θ = 50o and ϕ = 35o. The dimensional of signal plus noise subspace is only one. If a three element subset is chosen from this array, the dimension of the noise only subspace in V1N and V2N is two. We should be able to construct two independent Equations to define the signal’s DOA. The two different three element subsets consist of elements (1, 4, 7) and (1, 6, 7) respectively as given in Figure 4.
β k, j = −
2π sinθ j (x k cosφ j + y k sinφ j ) λ
(13)
In Equation.13, there are two unknown parameters, elevation angle θ and azimuth angle ϕ, that need to be determined. Thus, to obtain the DOA angles, two independent polynomials must be constructed and solved. There are several different techniques to derive the two independent polynomials.
z −1 w −1
6
5
The first approach constructs the two independent Equations from two distinct subsets [5]. Two distinct null spaces V1N and V2N can be derived from two different subsets. The two independent Equations are: H
(14)
H
(15)
J1(z, w) = aH(z,w)V1N V1N a(z,w) J2(z, w) = bH(z,w)V2N V2N b(z,w) π j sinθcosφ 2
j
3π sinθsinφ 2
where variables z = e ,w= e . Vectors a and b depend on the subset configurations. To guarantee the two Equations are independent, the two subsets cannot be related to each other by a linear shifting relation. The configuration of the subarrays has an affect on the direction finding accuracy of this algorithm; therefore for the best performance, the aperture of each subarray should be as large as possible.
ISBN: 978-988-17012-1-3
1
4
6
7
z −2
z −1 w
1
zw−1
z −1w−1
zw −1
2
3 zw
z2
5
z −2
7
1
4 z −1 w
1
2
z2
3 zw
Figure 4 Three Elements Subset Array Phase as a Function of z and w
The corresponding vector a and b would be a = [1 z-1w zw-1]T
(17)
b = [1 z-1w-1 zw-1]T
(18)
The z, w roots from the Equations J1(z, w) and J2(z, w) are shown in Figure 5. This result is derived when the SNR = 20 dB, and matrices V1N, V2N are constructed based on 100 snapshots of the observed data.
IMECS 2008
Proceedings of the International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, 19-21 March, 2008, Hong Kong 1
Z - Roots Estimated Signal Phasors
W - Roots Estimated Signal Phasors
1
0.8
Increasing the subset from 3 elements to 4 elements improves the estimation accuracy. Suppose the elements of the two subsets are (1, 2, 6, 7) and (1, 4, 5, 7) as shown in Figure 8, then the corresponding vectors a and b are:
0.8
0.6
0.6
0.4
0.4
0.2
0.2 0
0
-0.2
-0.2
-0.4 -0.4
-0.6 -0.6
-0.8
a = [1 z2 z-1w-1 zw-1]T
(22)
b = [1 z-1w z—2 zw-1]T
(23)
-0.8
-1 -1 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
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0.8
1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Figure 5 Roots of J1(z, w) and J2(z, w) for SNR = 20 dB
If the SNR reduces to 5 dB, the roots of polynomials J1(z, w) and J2(z, w) are slightly different from roots shown in Figure 5. Their roots are shown in Figure 6. Z - Roots Estimated Signal Phasors
1
zw−1
6
5 z
W - Roots Estimated Signal Phasors
1
z −1w−1
z −1w−1
6
7
−2
1
zw−1
1
z −2 5
z2
2
7
1
1
2
z2
0.8 0.6
0.5 0.4
4 z −1w
0.2
0
0
4 z −1w
3 zw
3 zw
-0.2 -0.4
-0.5
Figure 8 Four Elements Subset Array Phase as a Function of z and w
-0.6 -0.8
-1
-1
0
0.5
1
-1
-0.5
0
0.5
1
Figure 6 Roots of J1(z, w) and J2(z, w) for SNR = 5 d
The pair of roots closest to the unit sphere corresponding to the signal’s DOA [6]. Once this pair of roots is identified, the elevation angle and azimuth angle can be computed by the following Equations.
⎛ 2
⎞ 1 arg(w) ⎟⎟ ⎝ 3π sinφ ⎠
θ = sin −1 ⎜ ⎜
(20)
where
5 dB 53
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(21)
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55
20 dB
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51
51 theta
53
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49
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48
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47
Ave Error=0.2061 Variance=0.0193
46
33
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35 phi
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39
Ave Error=1.1873 Variance=0.6021
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40
45 30
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34
35 phi
36
37
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39
40
Figure 7 Estimated Signal’s DOA Based on 1000 Independent Simulations (a) SNR = 20 dB, (b) SNR = 5 dB
ISBN: 978-988-17012-1-3
47
Ave Error=0.1381 Variance=0.0081
45 30
31
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34
35 phi
36
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39
Ave Error=0.7739 Variance=0.2206
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40
45 30
31
32
33
34
35 phi
36
37
38
39
40
Figure 9 Estimated Signal’s DOA Based on 1000 Independent Simulations (a) SNR = 20 dB, (b) SNR = 5 dB
Comparing Figures 7 and 9, an increase subset size is seen to improve the estimation accuracy. By choosing a larger subset, the order of polynomials J1(z, w) and J2(z, w) become larger. Thus finding the roots of those polynomials requires more computation resources.
5 dB
54
53
50
49
46
Simulation results based on 1000 independent simulations using 3 element subsets at two different SNRs are shown in Figure 7. The variance and average error both increase with increased noise power as expected. The averaged estimation error for SNR = 20 dB and SNR = 5 dB are 0.206o and 1.187o respectively. The estimation variances are 0.019 and 0.602.
theta
54
20 dB
47
arg(u) = tan −1 ⎛⎜ Im(u) ⎞⎟ . ⎜ Re(u) ⎟ ⎝ ⎠
32
55
54
53
ϕ = cos ⎛⎜ 2 1 arg(z) ⎞⎟ ⎝ π sinθ ⎠
31
55
(19)
-1
45 30
The estimated signal’s DOA based on 1000 independent simulation using 4 element subsets at 2 different SNR is shown in Figure 9. The variance and average error both increase with increased noise power as expected. The averaged estimation errors for SNR = 20 dB and SNR = 5 dB are 0.138o and 0.774o respectively. The estimation variances are 0.008 and 0.221.
theta
-0.5
theta
-1
There are many different ways to pick the subsets from this seven element array. The 3 element subsets in the previous simulation are (1, 4, 7) and (1, 6, 7) respectively. If we choose different subsets consist of elements (1, 3, 6) and (1, 4, 7), the estimation variance is considerably smaller. The comparison of scatter diagram using different subsets based on 1000 independent simulations is shown in Figure 10. This new subsets reduces the estimation variance from 0.019 to 0.004, better than 6.7 dB improvement.
IMECS 2008
Proceedings of the International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, 19-21 March, 2008, Hong Kong 20 dB SNR 55
Subset Subset
54
Subset
54
53
Subset
53
52
52
51
51 theta
theta
beamwidth is about 57o. This antenna will not be able to resolve multiple targets if their angle separation is less than 57o. This limitation can be easily resolved by using array antenna and PRIME processing algorithm.
20 dB SNR
55
50
49
Suppose there are two signals impinging on this array with DOA (θ = 60o, φ = 15o) and (θ = 50o, φ = 35o), the angle separation between those two target is 15.4o. If the SNR is 20 dB, the estimated DOA based on 1000 independent trials by using 4 element subsets with element (1, 2, 6, 7) and (1, 4, 5, 7) is shown in Figure 13.
50
49
48
48
47
47
Ave Error=0.2061 Variance=0.0193
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45 30
31
32
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34
35 phi
36
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39
Ave Error=0.1162 Variance=0.0041
46
45 30
40
31
32
33
34
35 phi
36
37
38
39
40
Figure 10 Scatter Diagram for SNR = 20 dB using Subsets (a) (1, 4, 7) and (1, 6, 7), (b) (1, 3, 6) and (1, 4, 7) Signal 1 Ave Error=0.1074 Variance=0.0068
61
60.5
Actual Signal Direction Signal 1: θ = 600, φ = 150 Signal 2: θ = 500, φ = 350 Separation Angle=15.40
60
65 59.5
59 14
14.5
15
15.5
16
60
51 theta
Similar performance improvement is also observed in 4 element subsets. Figure 11 shows the scatter diagrams using different subsets based on 1000 independent simulations. The subsets used are (1, 2, 6, 7), (1, 4, 5, 7) and (1, 2, 3, 6), (1, 4, 5, 7) respectively. The SNR is assumed 20 dB.
50.5
55
50
20 dB SNR
20 dB SNR
Subset Subset
Subset Subset
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51 theta
theta
54
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49 34
34.5
35
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45 10
36
15
20
25
30
35
40
phi
Figure 13 Estimated DOA using 4 Element Subsets, Signals’ Angle Spacing Approximately equal One Quarter of Mainlobe Beamwidth
50
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48
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Signal 2 Ave Error=0.0784 Variance=0.0031
49.5
55
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48
Ave Error=0.1381 Variance=0.0081
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45 30
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35 phi
36
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39
47
Ave Error=0.0833 Variance=0.0032
46 40
45 30
31
32
33
34
35 phi
36
37
38
39
40
Figure 11 Scatter Diagram for SNR = 20 dB using Subsets (a) (1, 2, 6, 7) and (1, 4, 5, 7) , (b) (1, 2, 3, 6) and (1, 4, 5, 7)
Suppose the DOA of two signals are (θ = 50o, φ = 25o) and (θ = 50o, φ = 35o), the angle separation between those two targets is 8.4o. Under the same SNR and using identical subsets, the estimated signal DOA is shown in Figure 14. Actual Signal Direction Signal 1: θ = 500, φ = 250 Signal 2: θ = 500, φ = 350 Separation Angle = 8.40
51
By using new pair of subsets, the estimation variance is reduced from 0.008 to 0.003; the improvement factor is better than 4 dB.
Signal 1 Ave Error=0.0980 Variance=0.0050
50.5
50
55 54
49.5
53
100 Snap
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51 theta
theta
53
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Ave Error=0.7739 Variance=0.2206
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45 30
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40
Ave Error=0.2564 Variance=0.0226
46
45 30
31
32
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34
35 phi
36
37
38
39
40
Figure 12 Scatter Diagram for SNR = 5 dB (a) 100 Snapshots, (b) 1000 Snapshots
Figure 12 shows increasing the number of snapshots by a factor of 10, the estimation variance reduces from 0.2206 to 0.0226. A factor of 10 dB improvement is achieved. The aperture of a seven element antenna is very small. If this is a conventional fixed antenna, then the mainlobe
ISBN: 978-988-17012-1-3
25
25.5
26
51 50.5
50 49
50
48
Signal 2 Ave Error=0.1102 Variance=0.0083
49.5
49 34
34.5
35
35.5
36
47 46 45 20
25
30
35
40
phi
Figure 14 Estimated DOA using 4 Element Subsets, Signals’ Angle Spacing Approximately equal One Seventh of the Mainlobe Beamwidth
5 dB
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53
20 dB
52 24.5
1000 Snap
55
5 dB
54
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theta
Increasing the number of snapshots improves the estimation of the correlation matrix. If the elements of the two subsets are (1, 2, 6, 7) and (1, 4, 5, 7), the SNR = 5 dB, the estimated angles based on 100 and 1000 snapshots are shown in Figure 12.
Figures 13 and 14 show that with the PRIME processing algorithm, the array antenna can resolve multiple targets even their angle separation is considerably less than the conventional antenna mainlobe beamwidth. B. Sensitivity Analysis For a sensor array with a large number of elements arranged symmetrically on an x-y plane, the DOA estimation variance should be weakly depended on the azimuth angle. In this section, we use computer simulation to evaluate the estimation variance as a function of elevation angle due to noise, element position and phase error.
IMECS 2008
Proceedings of the International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, 19-21 March, 2008, Hong Kong Since element position error yields an equivalent phase error, the system imperfection can be lumped into a total equivalent phase error. Figure 15 shows the DOA estimation variance of seven element array when a single signal is impinging the array from different elevation angles. The signal to noise ratio is assumed to be 20 dB. This result is based on using two different three element subsets. Variance computations are derived from 100 independent simulations.
The PRIME algorithm studied in this paper derives the two independent polynomials J1(z, w) and J2(z, w) from two distinct subsets of the original array. Another possible approach is to derive two independent polynomials by using the full array, but in this case the null space matrices V1N and V2N should have at least one different column. We propose to investigate this approach in future studies.
ACKNOWLEDGMENT
Estimated Elevation (θ) Variance using PRIME with 3-Element Subset Array Configuratio 0.1
The authors would like to thank the Raytheon Corporation for its support of this investigation.
0.09 Estimated Elevation Variance
0.08 0.07
REFERENCES
0.06
[1]
0.05 0.04
[2]
0.03 0.02
[3]
0.01 0
0
10
20
30 40 50 60 Elevation (θ) Degrees
70
80
90
Figure 15 Estimation Variance vs Signal’s Elevation Angle
Figure 15 shows that when the signal’s elevation angle exceeds 75o, the estimation variance increases rapidly.
III. CONCLUSION
[4] [5]
[6]
Lee Anne, Chen Lijia, Song Aijin, Wei Jerone and Hwang H. K. “Simulation Study of Wideband Interference Rejection using Adaptive Array Antenna” IEEE Aerospace Conference, 2005 M. G. M. Hussain, “Performance Analysis and Advancement of Self-Steering Arrays for Nonsinusoidal Waves”, IEEE Trans. on Electromagnetic Compatibility, March 1988 X. Zhang and D. Su, “Digital Processing System for Digital Beam Forming Antenna” IEEE International Symposium on Microwave, Antenna Propagation and EMC Technologies, 2005 J. R. Guerci, “Space Time Adaptive Processing for Radar, Artech House Publisher, 2003 Hatke, Gary, and Keith Forsythe “A Class of Polynomial Rooting Algorithms for Joint Azimuth/Elevation Estimation Using Multidimensional Arrays.” 28th Asilomar Conference on Signals, Systems and Computers, p 694–699, 1994. Forsythe, Keith. “Utilizing Waveform Features for Adaptive Beamforming and Direction Finding with Narrowband Signals.” Lincoln Laboratory Journal 10.2 (1997): 99-126
This paper investigates the possibility of combining the array antenna and advanced signal processing techniques to enhance the estimation of the direction of signal sources. A conventional method to detect the direction of signal source is to use a fixed antenna to scan over certain searching region. Whenever there is a high received power from a particular direction, then we assume that is the signal’s DOA. This primitive estimation technique has many limitations. First its resolution is limited by the antenna mainlobe beamwidth. For small aperture antennas such as a missile seeker antenna, the resolution is very poor. Also, if there are multiple signal sources, a conventional fixed antenna has difficulty in detecting them simultaneously. Using the advanced signal processing techniques, the DOA estimation can be improved and one of the important algorithms is the PRIME algorithm. PRIME algorithms results are summarized as follows: 1. The performance of the PRIME algorithm improves if the subsets contain the largest number of possible elements, or the number of snapshots is increased. 2. For the subsets using the same number of elements, the particular subsets whose elements span the largest region has the best performance. 3. The array antenna can resolve multiple targets even though their angle separation is considerably less than the conventional antenna mainlobe beamwidth.
ISBN: 978-988-17012-1-3
IMECS 2008
